Field
For the field in relations, see field (relation). A field is a set paired with two operations on the set, which are designated as addition (+) and multiplication (\cdot) . As a group can be conceptualized as an ordered pair of a set and an operation, (G,\cdot) , a field can be conceptualized as an ordered triple (F,+,\cdot) . A set with addition and multiplication, (F,+,\cdot) , is a field if and only if it satisfies the following properties: #Commutativity of both addition and multiplication — For all a,b\in F , a+b=b+a and a\cdot b=b\cdot a #Associativity of both addition and multiplication — For all a,b,c\in F , (a+b)+c=a+(b+c) and (a\cdot b)\cdot c=a\cdot(b\cdot c) #Additive Identity — There exists a "zero" element, 0\in F , called an additive identity, such that a+0=a for all a\in F #Additive Inverses — For each a\in F , there exists a b\in F , called an additive inverse of a , such that a+b=0 #Multiplicative Identity — There exists a "one" element, 1\in F , different from 0, called a multiplicative identity, such that a\cdot1=a for all a\in F #Multiplicative Inverses — For each a\in F , except for 0, there exists a c\in F , called a multiplicative inverse of a , such that a\cdot c=1 #Distributive property — For all a,b,c\in F , a\cdot(b+c)=a\cdot b+a\cdot c #Closure of addition and multiplication — For all a,b\in F , a+b\in F and a\cdot b\in F Alternatively, a field can be defined as a commutative ring with unity (has a multiplicative identity) and multiplicative inverses. We will often abbreviate the multiplication of two elements, a\cdot b , by juxtaposition of the elements, ab . Also, when combining multiplication and addition in an expression, multiplication takes precedence over addition unless the addition is enclosed in parenthesis. That is, a+bc+d=a+(bc)+d . We can also denote -a and a^{-1} as additive and multiplicative inverses of any a\in F . Furthermore, we can define two more operations, called subtraction and division by a-b=a+(-b) , and provided that b\ne0 , \frac{a}{b}=a\cdot b^{-1} . Important Results Because a field is also a ring with unity, these properties are inherited: * (F,+) is an abelian groups * 0\cdot a=0 , for all a\in F * a(-b)=(-a)b=-(ab) , for all a,b\in F * (-a)(-b)=ab , for all a,b\in F * (-1)\cdot a=-a , for all a\in F * (-1)\cdot(-1)=1 *Multiplication distributes over subtraction. Additionally: * (F^*,\cdot) is also an abelian group, where F^* is the set of nonzero elements of F *Any field contains a subfield K that is field-isomorphic to \Q or \Z_p for some prime p . Optional Properties A field (F,+,\cdot) is: *A subfield of a field (K,+,\cdot) if F\subseteq K (see subset), where addition and multiplication on F is a domain restriction on the addition and multiplication on K . More commonly, we say that K is an extension field of F , and in fact, is also a vector space over F *An ordered field if there exists a total order \le on F such that for all a,b,c\in G , if a\le b , then a+c\le b+c , (translation invariance), and if 0\le a and 0 \le b , then 0\le ab Examples *Under the usual operations of addition and multiplication, the rational numbers ( \Q ), algebraic numbers ( \mathbb A ), real numbers ( \R ), and complex numbers ( \C ) are fields. *An extension field of \Q , such as \Q\left\sqrt2\right=\left\{a+b\sqrt2\mid a,b\in\Q\right\} . Related Elements of a field are the quantities over the vectorspaces are constructed and there are also called the scalars. In the same branch functions X\to F , where F is a field are called scalar fields. Category:Algebra Category:Linear algebra Category:Tensors Category:Differential geometry Category:Abstract algebra